C₂³×#62842 #72324522

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁵

Aliases

direct product

C₂³×#62842

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

16771240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 9 generators on 26 points:

(11,12)(13,18)(14,15)(16,17)(19,24)(20,26)(21,23)(22,25);(12,17)(14,19)(18,25)(20,23);(3,4);(5,6);(14,19)(15,24)(20,23)(21,26);(1,2)(3,4)(5,6);(8,10)(11,13)(12,18)(14,23)(15,21)(16,22)(17,25)(19,20)(24,26);(7,8)(9,10)(11,14)(12,15)(13,20)(16,19)(17,24)(18,26)(21,25)(22,23);(7,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,24)(18,20)(22,26)(23,25)

Conjugacy classes & character table

656 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₂³×#7631286, C₂³×#7918389, C₂²×#72235095, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((11,12)(13,18)(14,15)(16,17)(19,24)(20,26)(21,23)(22,25), (12,17)(14,19)(18,25)(20,23), (3,4), (5,6), (14,19)(15,24)(20,23)(21,26), (1,2)(3,4)(5,6), (8,10)(11,13)(12,18)(14,23)(15,21)(16,22)(17,25)(19,20)(24,26), (7,8)(9,10)(11,14)(12,15)(13,20)(16,19)(17,24)(18,26)(21,25)(22,23), (7,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,24)(18,20)(22,26)(23,25));

C23×#62842 #72324522